University of North Carolina at Chapel Hill Using Stata Graphics as a Method of Understanding and Presenting Interaction Effects Joanne M. Garrett, PhD University of North Carolina at Chapel Hill July 11, 2005

Problems with Understanding Interaction Interaction difficult concept to explain not a single answer (point estimate) linear combination of betas Often ignored because of difficulty Graph of interaction may be more intuitive for: students learning the concept presentations at professional meetings

Low Birth Weight Study* Variable Description Coding low Low birth weigh 1 = ≤ 2500 gms 0 = >2500 gms bwt Birth weight grams smk Smoked during pregnancy 1 = smoked 0 = did not smoke race Mother’s race 1 = black 0 = white age Mother’s age years * “Loosely” adapted from data from Applied Logistic Regression, David W. Hosmer, Jr. and Stanley Lemeshow

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) Logistic regression model: Add smk by race interaction Create interaction term and run logistic regression: . gen smkxrace = smk * race . logistic low smk race smkxrace

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) -------------------------------------------------- low | OR z P>|z| [95% CI] ---------+---------------------------------------- smk | 5.76 4.14 0.000 2.51 13.19 race | 5.43 4.13 0.000 2.43 12.15 smkxrace | .320 -2.08 0.038 .109 .936 Significant interaction: Relationship differs between smoking and low birth wt depending on mother’s race Question: How to interpret the interaction OR?

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) Convert OR’s to beta coefficients and solve by categories of race: . logit -------------------------------------------------- low | Coef. z P>|z| [95% CI] ---------+---------------------------------------- smk | 1.751 4.14 0.000 0.921 2.580 race | 1.693 4.13 0.000 0.889 2.497 smkxrace | -1.141 -2.08 0.038 -2.216 -0.066 _cons| -2.303 White: OR = e [β1 + β3(race)] = e [1.751 – 1.141(0)] = 5.76 Black: OR = e [β1 + β3(race)] = e [1.751 – 1.141(1)] = 1.84

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) White: . lincom smk + 0*smkxrace ---------------------------------------------- low | OR z P>|z| [95% CI] -----+---------------------------------------- (1) | 5.76 4.14 0.000 2.51 13.2 Black: . lincom smk + 1*smkxrace (1) | 1.84 1.75 0.081 .928 3.65

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) Interpretation: Among whites, women who smoke have 5.8 times the odds of having a low birth wt baby Among blacks, there is no relationship between smoking and having a low birth wt baby (OR=1.8, but not statistically significant)

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) Misinterpretations: Black mothers have less “risk” for low birth wt babies compared to white mothers It’s okay for black mothers to smoke Alternative: Solve the equation for values of smk and race Graph the individual probabilities (“predxcat”) . predxcat low, xvar(race smk) graph bar

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white) +-------------------------------------------------+ | race smk numobs prob lower upper | |-------------------------------------------------| | 0:White 0:No 88 0.091 0.046 0.171 | | 0:White 1:Yes 104 0.365 0.279 0.462 | | 1:NonWh 0:No 142 0.352 0.278 0.434 | | 1:NonWh 1:Yes 44 0.500 0.356 0.644 | Likelihood ratio test of interaction for race * smk: LR Chi2(1) = 4.56 Prob > Chi2 = 0.0328

Example 1: Low birth weight and interaction between smoking and race (1=black, 0=white)

Example 2: Low birth weight and interaction between smoking and age (years) Logistic regression model: Add smk by age interaction Create interaction term and run logistic regression: . gen smkxage = smk * age . logistic low smk age smkxage

Example 2: Low birth weight and interaction between smoking and age (years) ------------------------------------------------- low | OR z P>|z| [95% CI] ---------+--------------------------------------- smk | 0.382 -0.90 0.367 .047 3.110 age | 0.920 -2.61 0.009 .865 0.980 smkxage | 1.076 1.97 0.049 1.001 1.157 Significant interaction: Relationship differs between smoking and low birth wt depending on mother’s age Question: How to interpret the interaction OR for a continuous interaction variable?

Example 2: Low birth weight and interaction between smoking and age (years) Convert OR’s to beta coefficients and solve for selected values of age: . logit ------------------------------------------------- low | Coef. z P>|z| [95% CI] --------+---------------------------------------- smk | -.963 -0.90 0.367 -3.058 1.131 age | -.083 -2.61 0.009 -0.145 -0.021 smkxage | .073 1.97 0.049 0.001 0.146 _cons| .805 Age=15: OR = e [β1 + β3(age)] = e [–0.963 + 0.073(15)] = 1.14 Age=35: OR = e [β1 + β3(age)] = e [–0.963 + 0.073(35)] = 4.93

Example 2: Low birth weight and interaction between smoking and age (years) Age=15: . lincom smk + 15*smkxage ----------------------------------------------- low | OR z P>|z| [95% CI] -----+----------------------------------------- (1) | 1.14 0.32 0.751 .503 2.59 Age=35: . lincom smk + 35*smkxage low | OR z P>|z| [95% CI] (1) | 4.93 2.59 0.010 1.47 16.5

Example 2: Low birth weight and interaction between smoking and age (years) Interpretation: Among 15 year olds, women who smoke have 1.1 times the odds of having a low birth wt baby Among 35 year olds, women who smoke have 4.9 times the odds of having a low birth wt baby

Example 2: Low birth weight and interaction between smoking and age (years) Misinterpretations: 15 year olds are not at risk for low birth wt babies It’s okay for 15 year olds to smoke Alternative: Solve the equation and graph the probabilities for different levels of smoke and age (“predxcon”) . predxcon low, xvar(age) from(15) to(35) inc(2) class(smk) graph

Example 2: Low birth weight and interaction between smoking and age (years) -> smk = 0 +---------------------------------+ | age pred_y lower upper | |---------------------------------| | 15 .392 .272 .527 | | 17 .353 .259 .461 | | 19 .317 .243 .400 | | (etc) ... ... ... | -> smk = 1 | 15 .424 .285 .577 | | 17 .419 .303 .546 | Likelihood ratio test for interaction of age * smk: LR Chi2(1) = 3.88 Prob > Chi2 = 0.05

Example 2: Low birth weight and interaction between smoking and age (years)

Example 3: Birth weight (grams) and interaction between smoking and race Linear regression model: Create interaction term and run linear regression: . gen smkxrace = smk * race . regress low smk race smkxrace Or: . predxcat bwt, xvar(race smk) graph bar

Example 3: Birth weight (grams) and interaction between smoking and race

Example 4: Birth weight (grams) and interaction between smoking and age Linear regression model: Create interaction term and run linear regression: . gen smkxage = smk * age . regress low smk age smkxage Or: . predxcon bwt, xvar(age) from(15) to(35) inc(2) class(smk) graph

Example 4: Birth weight (grams) and interaction between smoking and age

Example 5: Birth weight (grams) and interaction between smoking and age, age2, age3 Linear regression model: add quadratic & cubic terms Create interaction terms and run linear regression: . gen age2 = age^2 . gen age3 = age^3 . gen smkxage = smk * age . gen smkxage2 = smk * age2 . gen smkxage3 = smk * age3 . regress low smk age age2 age3 smkxage smkxage2 smkxage3 Or: . predxcon bwt, xvar(age) from(15) to(35) inc(2) class(smk) graph poly(3)

Example 5: Birth weight (grams) and interaction between smoking and age, age2, age3

Conclusions Interaction can be a difficult concept for people unfamiliar with the methodology Examining a graph of an interaction is an easier way to get an intuitive feel for the effect A useful technique for explaining interaction to students hearing it for the first time, before introducing mathematical models A simple way to present study results at meetings, even to a statistically savvy audience

Calculating and Graphing Predicted Values: (when “X” is categorical) . predxcat yvar, xvar(xvar1 xvar2) yvar – dependent variable continuous – defaults to linear regression binary (0,1) – defaults to logistic regression xvar(xvar) – nominal variable for categories of estimated means or proportions xvar(xvar1 xvar2) – categories of all combinations of xvar1 and xvar2; tests interaction adjust(cov_list) – adjusts for any covariates

Calculating and Graphing Predicted Values: (when “X” is categorical) graph – display graph (otherwise shows list of predicted values only) bar – bar graph (instead of symbols – default) model – for display purposes only; displays regression model Some other options: level(#) cluster(cluster_id) savepred(ds_name)

Calculating and Graphing Predicted Values: (when “X” is continuous) . predxcon yvar, xvar(xvar) from(#) to(#) inc(#) graph yvar – dependent variable continuous – defaults to linear regression binary (0,1) – defaults to logistic regression xvar(xvar) – continuous independent variable; probabilities calculated for each value of X from(#) – bottom value for xvar to(#) – top value for xvar inc(#) – increment desired between bottom and top values adjust(cov_list) – adjusts for any covariates

Calculating and Graphing Predicted Values: (when “X” is continuous) graph – display graph (otherwise shows list of predicted values only) class(class_var) – adds an xvar by class_var interaction term poly(2 or 3) – polynomial terms added: 2=squared 3=squared and cubic model – for display purposes only; displays regression model Some other options: level(#) cluster(cluster_id) nolist savepred(ds_name)